Remarks on the existence of solutions to some quasilinear elliptic problems involving the Hardy-Leray potential

摘要

We study the solvability of the quasilinear problem ??_pu = u~q/|x|~p + g(λ, x, u) u >0 in?, with u = 0 on ??, where ??_p(·) is the p-Laplacian operator, q > 0, 1 < p < N and ? a smooth bounded domain in R~N. We consider the following cases: (i) g(λ, x, u) ≡ 0; (ii) g(λ, x, u) = λf (x)u~r, with λ > 0 and f (x) 0 belonging to L~∞(?) and 0 ≤ r < p ? 1. In the case (i), the existence of solutions depends on the location of the origin in the domain, on the geometry of the domain and on the exponent q. On the other hand, in the case (ii), the existence of solutions only depends on the position of the origin and on the coefficient λ, but does not depend either on the exponent q or on the geometry of ?.